Bank Runs and Investment Decisions Revisited
ABSTRACT In this paper we extend the Cooper and Ross (1998) analysis of the optimal response of a competitive bank to the possibility of a bank run. If the probability of a run is small, the bank will offer a contract that admits a bank-run equilibrium. We show that, in this case, the bank will hold a quantity of liquid assets large enough to exactly meet withdrawal demand if a run does not occur; "excess" liquidity will not be held. This result allows us to determine how the possibility of a bank run affects the level of long-term investment chosen by a bank. We show that when the cost of liquidating investment early is high, the level of investment is decreasing in the probability of a run. However, when liquidation costs are smaller, the level of investment is actually increasing in the probability of a run.
Bank Runs and Investment Decisions Revisited1
Research Department, Federal Reserve Bank of Richmond
Centro de Investigación Económica, ITAM
December 16, 2003
In this paper we extend the Cooper and Ross (1998) analysis of the optimal
response of a competitive bank to the possibility of a bank run. If the
probability of a run is small, the bank will offer a contract that admits a
bank-run equilibrium. We show that, in this case, the bank will hold a
quantity of liquid assets large enough to exactly meet withdrawal demand
if a run does not occur; “excess” liquidity will not be held. This result
allows us to determine how the possibility of a bank run affects the level
of long-term investment chosen by a bank. We show that when the cost of
liquidating investment early is high, the level of investment is decreasing in
the probability of a run. However, when liquidation costs are smaller, the
level of investment is actually increasing in the probability of a run.
expressed herein are those of the authors and do not necessarily reflect those of the Federal Reserve Bank of Richmond
or the Federal Reserve System.
Bank panics and banking crises in general have important macroeconomic effects. The direct
consequences of banking crises are now well documented.2However, the mere possibility of a run
on the banking system will also have important macroeconomic consequences, even if such a run
doesnotoccur. Forexample, thepossibilityofarunchangestheexpectedpayoffsassociatedwitha
given deposit contract, and hence will likely change the contract that a bank would choose to offer.
As a result, the consumption allocations available to consumers will be distorted. In addition, the
possibility of a run will likely change the portfolio choice made by a bank. One of the main roles
of the banking system is to perform maturity transformation; that is, to hold long-term assets while
issuing short-term liabilities. Hence an important question is to what extent the possibility of a
crisis will lead banks to hold fewer long-term assets and therefore provide less of this socially-
valuable function. These indirect effects of banking crises are less well understood, and are our
focus in this paper. Specifically, we ask: How does the possibility of a bank run influence the
deposit contract offered and the investment decisions made by a competitive bank?
We build on the work of Cooper and Ross , who first analyzed this issue in an extension of
the Diamond and Dybvig  model. Diamond and Dybvig  set up an environment with idiosyn-
cratic uncertainty and asymmetric information, and they study the optimal insurance arrangement
in that environment. They discover that the first-best allocation (under full information) is incen-
tive compatible. In other words, the first-best allocation is Nash implementable. However, they
also show that this allocation is not fully implementable in the sense that there exists another Nash
equilibrium of the direct revelation mechanism in which a proportion of the agents in the economy
misrepresent their type. This other equilibrium resembles a bank run where all agents attempt to
withdraw their funds from the contractual arrangement early, fearing that other agents will do the
same and that as a result there will be insufficient funds to fulfill all promised payments. Given
this multiplicity of equilibria, the issue of equilibrium selection becomes germane. A common
criticism of the Diamond and Dybvig  analysis is that if agents believe that the run equilibrium
will occur, then they will not accept the contract being offered to them.
Cooper and Ross  introduce a sunspots-based equilibrium selection rule to the problem: if
the banking contract is such that both equilibria exist, then the run equilibrium will be selected
See, for example, Caprio and Klingebiel  and Boyd et al. .
with a fixed probability q.3They restrict the bank to offer deposit contracts, in which depositors
are promised a fixed return and are given this return as long as the bank has not run out of funds.4
Banks operate in a perfectly competitive environment and hence choose the deposit contract that
maximizes the expected utility of depositors. Cooper and Ross  show that if the probability of
a run is above some cutoff level, the bank will set the deposit contract such that truth telling is a
dominant strategy for all depositors. Such a contract guarantees that a bank run will not occur, and
therefore is called run proof. However, they also show that if the probability of a bank run is below
the cutoff, the bank will choose a contract that is not run proof. In this case, bank runs emerge as a
truly equilibrium phenomenon; consumers are willing to deposit in the bank because the improved
one’s deposit if a run does occur.
Our interest is in characterizing the properties of the optimal deposit contract, and in examining
how this contract changes when the probability of a bank run changes. The Cooper-Ross model
has some attractive features for this purpose. First, the bank’s problem is conceptually easy to
the possibility of a run changes the bank’s incentives. Second, the problem has a small number
of choice variables, which allows us to give a relatively sharp characterization of the solution. In
spite of this apparent simplicity, our analysis yields some surprising results. In fact, we show that
some of the conclusions drawn in Cooper and Ross  are misleading, and our extension of their
results helps clarify these issues. Cooper and Ross  focus on the role of “excess liquidity,” that
is, liquid assets that the bank intends to hold over the long term, despite the fact that these assets
yield a lower rate of return than illiquid investment. There are in principle two reasons why a
bank might choose to hold excess liquidity. First, having a highly liquid portfolio would minimize
liquidation costs if a run occurs. Second, if the bank decides to offer a run-proof contract, holding
a highly liquid portfolio might provide the best possible run-proof consumption profile. Assuming
preferences are of the constant-relative-risk-aversion variety, we show that a bank will never hold
excess liquidity for the first reason. That is, if the bank chooses a deposit contract that is not run
proof, it will only hold enough liquid assets to exactly meet withdrawal demand if a run does not
if the bank is vulnerable to a run, a run will occur if and only if spots appear on the sun. The probability of spots
appearing on the sun is given by q, an exogenous parameter.
This restriction is designed to mimic a sequential service constraint; suspension of convertibility is not allowed.
In the problem considered by Diamond and Dybvig , the restriction to deposit contracts was non-binding since the
first-best allocation was being implemented. Here, however, the constraint is binding. Nevertheless, Peck and Shell
 show that the same qualitative results can obtain when a broad class of possible contracts is considered.
In other words, all agents are able to observe whether or not there are spots on the sun, and all agents believe that
occur. The only reason, then, that a bank would hold excess liquidity is as a way of making the
contract immune to runs. We also provide an exact characterization of the situations when the best
run-proof contract does indeed involve holding excess liquidity.
We then examine the relationship between the probability of a run and the level of illiquid
investment undertaken by the bank, maintaining our assumption about preferences. At first glance,
it seems like the nature of this relationship should be straightforward: the more likely is a run,
the more likely it is that the bank will have to liquidate all of its investment and therefore the less
that the bank should invest. However, we show that there is another, less obvious, effect at work.
If the bank chooses to hold a more liquid portfolio, it will also offer a higher return to agents
who withdraw their deposits early. This is an immediate implication of the result described above:
All liquid assets are used to meet short-term withdrawal demand. Because it is promising more
to each depositor, the bank will be able to serve fewer depositors in the event of a run. That is,
holding a more liquid portfolio leaves the bank with more resources if a run occurs, but it also
implies that the resources will be shared by fewer depositors (and more depositors will receive
nothing). We show that when liquidation costs are above some threshold, the first effect dominates
and investment is decreasing in the probability of a bank run. However, when liquidation costs are
more moderate, the second effect dominates and an increase in the probability of a run leads to an
increase in investment. This latter result is fairly counter-intuitive and has interesting implications.
If we think of banks as investing in productive capital, then economies where the probability of
a bank run is higher could have more investment in capital and hence experience faster growth in
periods where a run does not occur.
The remainder of the paper is organized as follows. In the next section, we briefly review the
environment of Cooper and Ross  and then set up the maximization problem of the bank. In
Section 3 we discuss the conditions under which the optimal contract is run proof, while Section
4 contains our results on the conditions under which the bank will choose to hold excess liquidity.
In Section 5 we examine the response of the level of investment to a change in the probability of a
bank run. Finally, in Section 6 we offer some concluding remarks.
2 The Bank’s Problem
The environment we study is exactly that of Cooper and Ross ; we follow their notation as
much as possible. Consider an economy with a [0,1] continuum of ex-ante identical agents, each
of whom lives for three periods. Each agent has an endowment (normalized to one) in period 0,
and no endowment in periods 1 and 2. In period 0, an agent decides whether or not to deposit her
endowment in the bank. With probability π ∈ (0,1), at the beginning of period 1 she will discover
that she is impatient and only gets utility from consuming in period 1. With probability (1 − π)
she will discover that she is patient and derives utility only from consuming in period 2. Let u(c)
be the utility over consumption (in the appropriate period) and assume that u is strictly increasing,
strictly concave, and satisfies u(0) = 0.
There are two technologies for saving, which we call storage and investment. One unit of
consumption placed in storage in either period 0 or period 1 yields one unit of consumption in
the following period. One unit of consumption placed in investment in period 0 yields R units of
consumption in period 2, but only (1 − τ) units if liquidated in period 1, where τ ≥ 0 represents a
We study the problem of a bank that behaves competitively in the sense that it offers the contract
that maximizes the expected utility of depositors. Following Cooper and Ross , we restrict
the bank to offer deposit contracts of the following form. There is a fixed payment cE that is
arriving in period 1 unless it has completely run out of funds; no suspension of convertibility
is allowed.6Whatever resources remain in the bank in period 2 are divided evenly among the
remaining depositors. Let cLdenote the payment promised to each of these depositors. In addition
to choosing these two payments, the bank must also decide how to divide the deposits it receives
between investment and storage. If a bank run does not occur in period 1, some of the resources
placed into storage may be left in the bank until period 2. These resources held in storage for two
periods represent “excess” liquidity. Let i denote the fraction of the bank’s deposits placed into
investment and 1 − i the fraction placed into storage. Define i2such that if a run does not occur,
the fraction 1 − i − i2of deposits will be paid out to depositors withdrawing in period 1. Then
i2represents excess liquidity as a fraction of total deposits, and iR + i2is the total amount of
consumption available for depositors arriving in period 2.
If a run were to occur in period 1, the total amount of resources available to the bank would
be equal to the sum of the funds held in storage and the amount of liquidity that can be obtained
choice is trivial. Cooper and Ross  introduced the liquidation cost in order to address issues related to portfolio
As Diamond and Dybvig  show, when the fraction of impatient consumers is known with certainty a sim-
ple suspension-of-convertibility scheme can costlessly eliminate bank runs. However, when this fraction is random
eliminating runs is costly and the optimal contract from a broad class that includes total and partial suspensions of
convertibility may permit runs to occur (see Peck and Shell).
Diamond and Dybvig assumedτ = 0, in which case storage is a dominated technology and the bank’s portfolio